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Chapter 1 Rotation of an Object About a Fixed Axis

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Chapter 1 Rotation of an Object About a Fixed Axis Chapter 1 Rotation of an Object About a Fixed Axis 1.1 The Important Stuff 1.1.1 Rigid Bodies; Rotation So far in our study of physics we have (with few exceptions) dealt with particles, objects whose spatial dimensions were unimportant for the questions we were asking. We now deal with the (elementary!) aspects of the motion of extended objects, objects whose dimensions are important. The objects that we deal with are those which maintain a rigid shape (the mass points maintain their relative positions) but which can change their orientation in space. They can have translational motion, in which their center of mass moves but also rotational motion, in which we can observe the changes in direction of a set of axes that is “glued to” the object. Such an object is known as a rigid body. We need only a small set of angles to describe the rotation of a rigid body. Still, the general motion of such an object can be quite complicated. Since this is such a complicated subject, we specialize further to the case where a line of points of the object is fixed and the object spins about a rotation axis fixed in space. When this happens, every individual point of the object will have a circular path, although the radius of that circle will depend on which mass point we are talking about. And the orientation of the object is completely specified by one variable, an angle θ which we can take to be the angle between some reference line “painted” on the object and the x axis (measured counter-clockwise, as usual). Because of the nice mathematical properties of expressing the measure of an angle in radians, we will usually express angles in radians all through our study of rotations; on occasion, though, we may have to convert to or from degrees or revolutions. Revolutions, degrees and radians are related by: 1 revolution = 360 degrees = 2π radians 1 2 CHAPTER 1. ROTATION OF AN OBJECT ABOUT A FIXED AXIS r s q Figure 1.1: A point on the rotating object is located a distance r from the axis; as the object rotates through an angle θ it moves a distance s. [Later, because of its importance, we will deal with the motion of a (round) object which rolls along a surface without slipping. This motion involves rotation and translation, but it is not much more complicated than rotation about a fixed axis.] 1.1.2 Angular Displacement As a rotating object moves through an angle θ from the starting position, a mass point on the object at radius r will move a distance s; s length of arc of a circle of radius r, subtended by the angle θ. When θ is in radians, these are related by s θ = θ in radians (1.1) r If we think about the consistency of the units in this equation, we see that since s and r both have units of length, θ is really dimensionless; but since we are assuming radian measure, we will often write “rad” next to our angles to keep this in mind. 1.1.3 Angular Velocity The angular position of a rotating changes with time; as with linear motion, we study the rate of change of θ with time t. If in a time period ∆t the object has rotated through an angular displacement ∆θ then we define the average angular velocity for that period as ∆θ ω = (1.2) ∆t A more interesting quantity is found as we let the time period ∆t be vanishingly small. This gives us the instantaneous angular velocity, ω: ∆θ dθ ω = lim = (1.3) ∆t→0 ∆t dt rad 1 −1 Angular velocity has units of s , or equivalently, s or s . 1.1. THE IMPORTANT STUFF 3 In more advanced studies of rotational motion, the angular velocity of a rotating object is defined in such a way that it is a vector quantity. For an object rotating counterclockwise about a fixed axis, this vector has magnitude ω and points outward along the axis of rotation. For our purposes, though, we will treat ω as a number which can be positive or negative, depending on the direction of rotation. 1.1.4 Angular Acceleration; Constant Angular Acceleration The rate at which the angular velocity changes is the angular acceleration of the object. If the object’s (instantaneous) angular velocity changes by ∆ω within a time period ∆t, then the average angular acceleration for this period is ∆ω α = (1.4) ∆t But as you might expect, much more interesting is the instantaneous angular accelera- tion, defined as ∆ω dω α = lim = (1.5) ∆t→0 ∆t dt We can derive simple equations for rotational motion if we know that α is constant. (Later we will see that this happens if the “torque” on the object is constant.) Then, if θ0 is the initial angular displacement, ω0 is the initial angular velocity and α is the constant angular acceleration, then we find: ω = ω0 + αt (1.6) 1 2 θ = θ0 + ω0t + 2 αt (1.7) 2 2 ω = ω0 + 2α(θ − θ0) (1.8) 1 θ = θ0 + 2 (ω0 + ω)t (1.9) where θ and ω are the angular displacements and velocity at time t. θ0 and ω0 are the values of the angle and angular velocity at t = 0. These equations have exactly the same form as the equations for one–dimensional linear motion given in Chapter 2 of Vol. 1. The correspondences of the variables are: x ↔ θ v ↔ ω a ↔ α . It is almost always simplest to set θ0 = 0 in these equations, so you will often see Eqs. 1.6—1.9 written with this substitution already made. 1.1.5 Relationship Between Angular and Linear Quantities As we wrote in Eq. 1.1, when a rotating object has an angular displacement ∆θ, then a point on the object at a radius r travels a distance s = rθ. This is a relation between the angular motion of the point and the “linear” motion of the point (though here “linear” is a bit of a misnomer because the point has a circular path). The distance of the point from the axis 4 CHAPTER 1. ROTATION OF AN OBJECT ABOUT A FIXED AXIS does not change, so taking the time derivative of this relation give the instantaneous speed of the particle as: ds dθ v = = r = rω (1.10) dt dt which we similarly call the point’s linear speed (or, tangential speed), vT ) to distinguish it from the angular speed. Note, all points on the rotating object have the same angular speed but their linear speeds depend on their distances from the axis. Similarly, the time derivative of the Eq. 1.10 gives the linear acceleration of the point: dv dω a = = r = rα (1.11) T dt dt Here it is essential to distinguish the tangential acceleration from the centripetal acceleration that we recall from our study of uniform circular motion. It is still true that a point on the wheel at radius r will have a centripetal acceleration given by: v2 (rω)2 a = = = rω2 (1.12) c r r These two components specify the acceleration vector of a point on a rotating object. (Of course, if α is zero, then aT = 0 and there is only a centripetal component.) 1.1.6 Rotational Kinetic Energy Because a rotating object is made of many mass points in motion, it has kinetic energy; but since each mass point has a different speed v our formula from translational particle motion, 1 2 K = 2mv no longer applies. If we label the mass points of the rotating object as mi, having individual (different!) linear speeds vi, then the total kinetic energy of the rotating object is 1 2 1 2 Krot = 2mivi = 2 mivi i I X X th If ri is the distance of the i mass point form the axis, then vi = riω and we then have: 1 2 1 2 2 1 2 2 Krot = 2 mi(riω) = 2 (miri )ω = 2 miri ω . i i i ! X X X 2 The sum i miri is called the moment of inertia for the rotating object (which we discuss further inP the next section), and usually denoted I. (It is also called the rotational inertia in some books.) It has units of kg · m2 in the SI system. With this simplification, our last equation becomes 1 2 Krot = 2Iω (1.13) 1.1. THE IMPORTANT STUFF 5 1.1.7 The Moment of Inertia; The Parallel Axis Theorem For a rotating object composed of many mass points, the moment of inertia I is given by 2 I = miri (1.14) Xi I has units of kg · m2 in the SI system, and as we use it in elementary physics, it is a scalar 1 (i.e. a single number which in fact is always positive). More frequently we deal with a rotating object which is a continuous distribution of mass, and for this case we have the more general expression I = r2dm (1.15) Z Here, the integral is performed over the volume of the object and at each point we evaluate r2, where r is the distance measured perpendicularly from the rotation axis. The evaluation of this integral for various of cases of interest is a common exercise in multi-variable calculus. In most of our problems we will only be using a few basic geometrical shapes, and the moments of inertia for these are given in Figure 1.2 and in Figure 1.3.
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